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Science Forum Index » Physics - Electromagnetic Forum » Paradox
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| Author |
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| aether22@gmail.com |
 Posted: Tue Feb 19, 2008 4:46 pm |
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The basis of the problem is this.
Common sense and coil calculators (and even real world experiuments I
did looooong ago) all assure me that 2 identical hoop coils where the
only difference is the diameter should produce very different magnetic
field strengths (densities) when an identical current is passed.
The tighter coil produces a higher gauss than the same number of
Ampere turns in a larger dia. coil.
The next fact that can't be denied is that a coil wound on a highly
magnetically permeable toroid should have no significant readily
measurible magnetic flux outside the toroid provided the core has not
been saturated. (I am well aware that infact strong magnetic fields do
exist outside the toroid but due to superposition they are not readily
detected by magnetic means)
These 2 facts collide in a seeming impossibility however, if you have
(to keep it easy) 2 square toroid forms (a normal E I bar transformer
with the middle of the E removed) and you wind a single layer coil of
say 100 turns on (the outside vertical leg of) each of these toroid
forms and pass a current through there should not be an observible
external magnetic field from either of these toroids. see fig 1.
If we now place the 2 coils next to each other (as in fig 2) we should
not expect any drop in the gauss in either core since neither produced
a net external magnetic field.
Now consider fig 3, we have a show down (largest gauss wins) between a
tight coil over one core or a larger looser coil (of the same number
of ampere turns) over 2 cores, the tighter coil should win, easy
right?
Now in fig 4 we have the same only we have another core we have placed
next to the tight coil, which one wins now?
Since we have alrewady established that the tight cores shouldn't
interfere with each other the 2 tight cores should win.
But by now surely you can see the paradox, the only difference between
the 2 tight coils and the one loose one is the windings between the 2
tight coiled cores, but these shouldn't create any net magnetic field,
fig 5 helps illistrate this issue.
Anyone wanna take a crack at this, either explaining the paradox or at
least telling me in each case if you agree with the winner I've
chosen.
Or run it through a 3D magnetic simulation program?
Or at least agree it's a head scratcher!?!?
figs here:
http://www.imagedump.com/index.cgi?pick=get&tp=527437
note: in my browser I need to scroll horizontally to the right to see
the picture.
note 2: I have found that diameter has little effect in longer coils
so the coils may need to be kept shorter than pictured. |
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| Timo A. Nieminen |
| Posted: Wed Feb 20, 2008 2:09 pm |
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Guest
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On Tue, 19 Feb 2008, aether22@gmail.com wrote:
Quote: The basis of the problem is this.
Common sense and coil calculators (and even real world experiuments I
did looooong ago) all assure me that 2 identical hoop coils where the
only difference is the diameter should produce very different magnetic
field strengths (densities) when an identical current is passed.
The tighter coil produces a higher gauss than the same number of
Ampere turns in a larger dia. coil.
Sure, a consequence of the geometry. Field due to an infinitesimal current
element falls off as 1/r^2, field due to an infinite straight wire falls
off as 1/r. Thus, from either of these, the field at the centre of the
coil falls off as 1/r, r now being the coil radius (length of coil * 1/r^2
in the first case, a loop, seen from the centre, being equivalent to an
infinite wire in the 2nd).
Quote: The next fact that can't be denied is that a coil wound on a highly
magnetically permeable toroid should have no significant readily
measurible magnetic flux outside the toroid provided the core has not
been saturated.
[cut]
Anyone wanna take a crack at this, either explaining the paradox or at
least telling me in each case if you agree with the winner I've
chosen.
So, a significantly different geometry. Basically, the field now looks
like that produced by an infinitely long solenoid, which only depends on
turns/metre and the current, not the radius of the winding.
There's a geometric analogy along the lines of infinite wire vs loop. Note
that the field due to an infinite sheet of current is uniform. Wrap it
into a loop, and the field inside in uniform.
In practice, you're not going to have an infinite current sheet, or the
equivalent infinite solenoid, but you can wrap it into a toroid or around
a permeable core.
While at it, consider the magnetic moment of a closed loop. If you make
the field within the loop uniform, how does the field depend on the loop
(or coil) radius?
But I don't need to tell you this, since you already note at least the
kernel of the answer:
Quote: note 2: I have found that diameter has little effect in longer coils
so the coils may need to be kept shorter than pictured.
--
Timo |
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